Formula for the expectation $E|X|=\int_{0}^{\infty} P[|X|>t] dt$

Asked Sep 24 '19 at 02:17

Active Sep 24 '19 at 02:17

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I wonder why $E|X|=\int_{0}^{\infty} P[|X|>t] dt$. X is a positive random variable with density $f_X$

What I have done so far $$\int_{0}^{\infty} P[|X|>t] dt = \int_{0}^{\infty} \int_{t}^{\infty} f_X (y) dy dt = \int_{0}^{\infty} \int_{0}^{y} f_X (y) dt dy $$

From there, I am not sure how to proceed.

Thanks in advance!

asked Sep 24 '19 at 02:17

Casper Lindberg

1

$|X|$ is a nonnegative random variable and so your question is a duplicate of Explain why $E(X) = \int_0^\infty (1-F_X (t)) , dt$ for every nonnegative random variable $X$ – Dilip Sarwate Sep 24 '19 at 02:22
And you already come to the point that you can integrate the inner integral. – BGM Sep 24 '19 at 02:38

Formula for the expectation $E|X|=\int_{0}^{\infty} P[|X|>t] dt$

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